Sensors Grouping Model for Wireless Sensor Network *

The grouping of sensors is a calculation method for partitioning the wireless sensor network into groups, each group consisting of a collection of sensors. A sensor can be an element of multiple groups. In the present paper, we will show a model to divide the wireless sensor network sensors into groups. These groups could communicate and work together in a cooperative way in order to save the time of routing and energy of WSN. In addition, we will present a way to show how to organize the sensors in groups and provide a combinatorial analysis of some issues related to the performance of the network.


Introduction
A wireless sensor network consists of spatially distributed autonomous sensors to monitor physical or environmental conditions, such as temperature, sound, pressure, etc. [1][2][3][4][5].The sensors cooperate with each other to monitor the targets and send the collected information to the base station [6][7][8].Sensors are battery-powered devices having a limited lifetime, restricted sensing range, and narrow communication range [9][10][11], and densely deployed in harsh environment [12][13][14][15].Organization of sensors in the form of groups is very important, which would facilitate transferring data and routing from one group to another, and it also offers an easy way to analyze the WSN problems such as coverage, localization, connectivity, tracing and data routing [16][17][18][19][20].

Sensors Grouping Strategy
A Group of sensors is a collection of overlapped sensors in a single area.Let us define the degree of overloaded sensors by the maximum number of sensors overlapped in the same area, here we denote to the maximum coverage degree of an area by  that occurs in the area of intersection, which means that there is one and only one area covered by two sensors and that is the maximal overlapping that could be produced, so sensor 2 s and 3 s create a group of two sensors denoted by   2  1 2 .Sake of convenience, we denote to the group of sensors that build up the WSN by which we call it the mother network group or simply the mother group.

Counting the Sub-Areas of Sensors Group
Here we start by asking, how many sub-areas are generated if unit disk sensors are partially overlapped?Assuming there is no fully overlapping between sensors, and all sensors are homogenous (sensors have the same sensing range).Say   F k is the function to count the number of sub-areas, definitely Proof: suppose we have a group of sensors as shown in Figure 3, we can see that the number of areas inside each sensor's range is seven.Using Top-down approach from 1 , , , G s s s s  s to 4 s , the number of areas for the top sensor 1   s is seven (red areas namely 1, 2, 3, 4, 5, 6, 7).The number of areas inside the sensor's range 2 s are seven, namely (4, 5, 6, 7, 8, 9 10), but the red colored areas, 2, 3, 4, 5, already counted in 1 s , so there are only 3 blue colored areas inside sensor's range 2 s , namely 8, 9, 10.For the sensor 3 s , it has seven areas inside its sensing range, the 3, 5, 6, 7 are red areas already counted in sensor's range 1 s , the area 10 is blue area already counted in sensor's range 2 s , thus still two black areas in sensor's range 3 s , namely 11, 12.For sensing range of 4 s , there are seven areas inside it, three are red areas (4,5,6), two are blue areas (9, 10), and one area is black (11), so there is still one area only in 4 s (13).Therefore, the number of areas from top to down is 7 + 3 + 2 + 1 = 13.Generally, counting the sub-areas from top to down, the top sensor contains In addition, we can proof theorem 1 by counting the areas of a group basing on the degree of coverage, if an area covered by k sensors then it called k-covered area.For a group k G there is only one area is k-covered (the maximum degree of coverage), in the remainder areas, there are k areas are1-covered, k areas are 2-coverd, k areas are 3-coverd… k areas are k-1 covered.Let k j  be the number of areas that j-covered inside a group of sensors k G .For example, Lemma 1: for sens there are only one area eas are 2-coverd, k co rage for eac tio a Sensors Gr  G , all sensors have the same characteristics, for example, the number of areas, the degree of cove n points located on the border of the sensor, and the number of intersection points located inside sensor's range.

Counting the Intersection Points of
Counting the intersection points of k-overlap is an easy combination problem.Before prov te to the number of in Proof: Assuming t sensor has two inte sensor, since each sensor has hat there are k sensors and each rsection points with each neighbor intersection points with others, applying this method to all sensors, we get the total amount of intersection points as However, while calculating, ev gle point has been repeatedly counted twice, thus the right answer in regards to the intersection points quantity should be  ery sin We can use top down approach to calculate the number of intersection points, as shown in the Figure 5(a)   . 1 s is the top sensor; 5 s is the bo range of the top sens ttom sensor of the group.The number of intersection points inside (internal) and on the border of sensing or s , there are k-2 of intersection points.In the third node   3 s , there are k-3 of intersection points.
In the fourth node  4  s , there section points, and th is 0 intersection points in the bottom sensor.
Totally the number ntersection points is Another method t n o count the intersectio points of a group, we can imagine that the number of intersection points as the number of 2-permutation of k sensors, for example, S is a set of overlapped sensors   , , , S s s s s  . The 2-permutation of S is: , , , , , , s s s s s s , , , , , , s s s s s s , , , , , , , , , , We can use the Recurrence relation to find the ber of intersection points of a group of sensors.We can find the recurrence relation of the number of intersection points as num below: which can be easily solved using generatio [21].(See the proof of theorem 3), the solution is n function

 That Located within the Sensing Rang of a Sensor Associated to a Group (Internal
In F f inter 5, in Figu section points lo-

Counting the Number of Intersection Points
Points and External Points) igure 5(c), we can see that when k = 3 the number o section points located in the black sensor are re 5(b) k = 4, the number of inter cated in the red sensor are 9, when k = 5 the number of intersection points are 14.
Theorem 3: for a group of sensors k G , the number of intersection points within the sensing range of sensor is Proof: it is easy to realize tha the num -t ber of intersec tion points (internal and external) of the sens is satisfying the recursive relation:  So finding the solution to this recursiv elation is the proof of the theorem.Su The external intersection points of sensors are the points located on the border of a sensor.However, the internal points are those points located inside the sensors but not on the border.
Lemma 3 (the number of external points): the number of intersection points located on the border of a sensor, which belongs to a group of sensors k G is Proof: from Figure 5, it is easy to realize that the number of intersection points (external) of the sensor is satisfying the recursive relation: We can solve this relation using generation function as in the proof of theorem 3. Therefore, the solution to this recursive relation is the proof of this theorem Lemma 4 (the number of internal points): the number of intersection points located inside a sensor (not including the points located on the border) is Proof: from Figure 5, it is easy to realize that the number of intersection points (internal ) of the sensor is satisfying the recursive relation:


We can solve this relation using generation function as in the proof of theorem 3. Therefore, the solution to this recursive relation is the proof of this theorem.

 
From lemma 3, and lemma 4, we get the number of intersection points of a sensor that belongs to a group of sensors G E s by counting the intersection points located border of the sensor (external points) and the intersection points located inside the s nternal po on the ensor (i ints).

Counting the Number of Areas within the Sensing Range of a Sensor That Belongs to a Group k G
Theorem 4: e number of areas inside the sensor Proof: it is easy to realize that the number of areas inside the sensor is satisfying the recursive relation: We can solve this relation using generation func (7) tion as in the proof of theorem 3. Therefore, the solution to this recursive relation is the proof of this theorem

Counting the Number of Areas Located within the Sensing Range of a Sensor That Belongs to Multiple Groups
In Figure 1, the network sensors group : is , , , , , , , , , , , , , where a, b, c are the positive integers , , , en the mother group of s r enso is , , , , , , , , , , , , G s s s s s s s  Since s G  , then the mother group of sensor 5 s is  , , , , Let us now count the number of areas inside a se or; these areas are generated by intersection of mu ple groups of sensors.For facilitate, let us define

G G 
Q s as the number of areas inside a sensor i s , sens 1 wh rated by overlapping of a group of ors explained above, the mother group of ich are gene , s , according to orem the number of areas which created inside The total number of areas inside a sen s , which, associated to multiple groups is denoted by these ar re created by intersection the mo oups .form the firs eas a of sensors belong ther gr t glance, the However, this form is not correct, because i s is an element belongs to every sup-group of * Below we can count the number of areas of sensors of Figure 7  

Number of Distributed Messages
One of our aims is to find the number of messages that will be generated during of sensor  To generalize this idea, we can write the equation further.
We have i s associated to the mother group , the order of mother group of s is as t quation belo he e w   Here the integer number  is the count of subgroups of * .In addition, c is the repetition.
Theorem 4: The number of distributed messages sent form i s , associated to a mother roup * The num essages depends on the degree overlapped sensors.The more the degrees of coverage are, the more the areas will be generated.T get moves with r ber of m of herefore, the more messages will be generated.When a tar in the range of a senso   1 s , it will send notificatio er ersection areas and a certain number of sensors cover these areas, n messages to all neighbors but certainly not to itself.Since the sensor contains a certain numb of int th [8] A. Mainwaring, J. Polastre, R. Szewczyk and D. Culler, e sensor will send a notification message to all the sensors that cover the same area.

 
 s are the overlapped sensors, and x the number of overlapped sensors.The overlapped sensors that create a degree of an area

Figure 2
Figure 2. (a) the number of sub-areas of (b) the number of sub-areas of (c) the number o -a as of 1 G f sub deno

2 :
s to a group k-covered, k area 1-coverd, areas are 3-coverd… and k areas k-1 h area, the number of intersec-for a group k

Figure 4
Figure 4. (a) group of sensors, (b) group of sensors.5 G

Figure 6 .Figure 7 .
Figure 6.The number of areas inside sensor by group of sensors represent the degree of coverage.